The application of Fourier techniques to pricing and hedging financial derivatives has flourished in computational and mathematical finance over the last decade because most of the underlying processes driving derivatives have a characteristic function (i.e., the Fourier transform of the probability density function (PDF)). In general, two Fourier techniques are applied to option pricing: fast Fourier transforms (FFTs) and Fourier series, such as Fourier cosine (COS) series. To explain the first technique, FFTs in option pricing, we first consider a European option pricing formula that is a risk-neutral expectation of the discounted payoff of the option driven by an underlying process. This formation naturally implies the integration of the discounted payoff and the PDF of the underlying process. As the underlying process has a characteristic function, one can apply an inverse continuous Fourier transform to recover the PDF, discretise the continuous Fourier integral and use an FFT algorithm that computes discrete Fourier transforms in \(O(N \log _2(N))\)  operations to obtain the option prices [1]}, [2]}, [3]}, [4]}, [5]}, [6]}. The main disadvantage of using these methods is that they require thousands of grid points and considerable computational time to reach an acceptable level of accuracy. Another approach is to use Fourier series to expand a PDF in terms of a partial sum of orthogonal basis functions. The basis function can be complex an exponential, e.g., [7]}, or COS, e.g. [8]}, function. Each coefficient in the series is approximated via the characteristic function of the PDF. By integrating each basis function with the discounted payoff function of the option, we can obtain an option price. This technique originated in [8]} to price European-style options with COS basis functions. The success of Fourier COS series in European option pricing has lead to the rapid expansion of pricing and hedging options with early-exercise features and exotic options, like Asian, multi-asset or barrier options [10]}, [11]}, [12]}, [13]}. Compare with FFTs, the advantage of using Fourier series for option pricing is that they can achieve a global spectral (exponential) convergence rate and require fewer summation terms as long as the governing PDF is sufficiently smooth. However, using any type of Fourier series to represent a \(\mathbf {C}^\nu \)A vector space in which functions are \(\nu \)  continuously differentiable. piecewise continuous (non-smooth) functionA function is called piecewise continuous on an interval if the function is made up of a finite number of \(\nu \)  times differentiable continuous pieces., e.g., a non-smooth PDF, is notoriously fraught. Discontinuities cause the Gibbs phenomenon, which has two important consequences for the Fourier partial sum of length \(N\) :
